I wish I knew how many exactly, I have just started researching this. Late in the war bounties for foreign soldiers is reported between $500 to $800 was offered by Northern agents. One agent bragged he was ready to send 20,000 Irishmen. Now the details are merky but this is a little of what I have so far.
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It was illegal for subjects of the Crown to enlist in foriegn armies according to the British Foriegn Enlistment Act. An act that goes, as far as I know, as far back as 1819 and has been modified many times since. I have found several cases of arrest of agents or persons trying to enlist men into the Union army.
examples:
"Colonel Rankin, who was a member of Parliament in the Province of Canada and a militia officer, set out to raise a regiment of sixteen hundred lancers for service with the North. He was quickly arrested for violating the British Foreign Enlistment Act."
http://www2.marianopolis.edu/quebechistory/frncdns/studies/dcb/2.htm
USS Kearsarge incident.
Edwin George Eastman(1833-1872). He was appointed Consul by U.S. President Abrahm Lincoln to Queenstown, Ireland and served from 1863 to mid 1869. He was involved in a diplomatic incident in November of 1863 where he was alleged by the British to have enlisted Irish men into the US Navy on board the war sloop "Kearsarge" while it was in Queenstown harbor.
About a dozen men were found aboard the Kearsarge and turned over to the authorities in Ireland, arrested and jailed. This incident required President Lincoln and Congress to produce resolutions that the United States was not actively recruiting on foreign soils.
In Belgium about two hundred men were recruited and there may have been as many as five thousand. I have found many letters from Confederate diplomats complaining and notifing heads of state of Europe of the illegal practices of Northern agents recruiting in their countries (Naval O.R.)
From the O.R. Naval
"CONFEDERATE STATES COMMISSION,
London, June 6, 1863.
SIR: * * * As pertaining to the subject of your last dispatch, it may be proper to observe that on an enquiry made in the House of Commons a few days since with respect to the alleged recruiting for the Federal Army going on in Ireland, the chief secretary for Ireland, Sir Robert Peel, stated that the Government were aware of the presence there of Federal agents, though, of course., they did not openly recruit; that a great number of young men had certainly gone from Ireland to America whose passage had been paid, and that they had every reason to suppose that many had been induced to enlist in the Federal Army; that the Government was following the matter up and endeavoring to come at the facts if possible.
I have the honor to be, with great respect, your obedient servant,
J. E. MACFARLAND,
Secretary of Special Comnii88ion at London."
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From the American Memory, Lincoln Papers.
A Letter.
"William Smyth to Abraham Lincoln, Saturday, January 07, 1865 (Support from Ireland)
Sir
I congratulate you on the recent successes I was uneasy about Gen'l Sherman. I see you want 300 thousand men to put the finishing stroke to the War giving a bounty of 800 dollars to each man. Some months ago it was expected the American Gov. intended granting a free immigration to Irish among other people, and it was reported I was an agent. I was actually overwhelmed with applications. A soldier of the Class?? position told me not many days ago that every man in the Regiment, 900 strong, would immigrate understanding the purpose if I paid their passages and give them some money to leave with their families. You see how freely I write because I know I am writing to a gentlemen. From your exalted position you could not of course do the same if you are not badly advised you will send me L20 (pounds) a head and I will send you 5000 men. A letter can be written by some person I no only saying "I enclose your draft or drafts on the Bank of Ireland, The Provisional Bank of Ireland, the National Bank of Ireland for 100 thousand pounds you know the instructions for ABC" and I pledge you my word as a gentleman the money shall be properly applied. It? would not answer for ???? or any of your consular agents to know anything about the matter. If Gen'l Grant has 5000 fresh men he would soon crush Lee and all war then over and President Lincoln immortalized and mark my words you may regret not taking my advice. As to the conscription no free state will submit to it and you will rather injure than save yourself by resorting to it. I know it by letters from Irish people in America to their friends in this country. If you have a war with England about Canada which is considered very likely, the English navy is strong enough to break the blockade and seriously damage your trade and immigration should cease so be wise which there is yet time. The Arming of the slaves by the Rebels will delay the termination of the war. The notes passed by the emigrants will be in America before them. They must pay or serve,?? the former they will not be able to do. "
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David Upton